Classical and quantum duality in jordanian quantizations
P. P. Kulish, V. D. Lyakhovsky
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9 pages, Latex, presented at the VII-th Colloquium "Quantum Groups and Integrable Systems",Prague 18 - 20 June 1998, to be published in Czech. Journal of Physics
The limiting transitions between different types of quantizations are studied by the deformation theory methods. We prove that for the first order coboundary deformation (g,g*_1 + x g*_2) of a Lie bialgebra (g,g*) one can always get the quantized Lie bialgebra (g,g*_2) as a limit of the sequence of quantizations of the type A(g,g*_1). The obtained results are illustrated by some low-dimensional examples of quantized Lie algebras and superalgebras.
The quantum deformation of the Jordanian twist F_qJ for the standard quantum Borel algebra U_q(B) is constructed. It gives the family U_qJ(B) of quantum algebras depending on parameters x and h. In a generic point these algebras represent the hybrid (standard-nonstandard) quantization. The quantum Jordanian twist can be applied to the standard quantization of any Kac-Moody algebra. The corresponding classical r-matrix is a linear combination of the Drinfeld- Jimbo and the Jordanian ones. The obtained two-parametric families of Hopf algebras are smooth and for the limit values of the parameters…
We construct explicit Drinfel'd twists of Jordanian type for the generalized Cartan type K Lie algebras in characteristic 0 and obtain the corresponding quantizations, especially their integral forms. By making modular reductions including modulo p and modulo p-restrictedness reduction, and base changes, we derive certain modular quantizations of the restricted universal enveloping algebra $\mathbf u(\mathbf{K}(2n{+}1;\underline{1}))$ for the restricted simple Lie algebra of Cartan type K in characteristic p. They are new pointed Hopf algebras of noncommutative and noncocommutative and with di…
We introduce and investigate a one parameter family of quantum spaces invariant under the left (right) coactions of the group-like element ${\cal T}_{h}^{(j=1)}$ of the Jordanian function algebra $Fun_{h}(SL(2))$. These spaces may be regarded as Jordanian quantization of the two-dimensional spheres.
Jordanian quantizations of Lie algebras are studied using the factorizable twists. For a restricted Borel subalgebras ${\bf B}^{\vee}$ of $sl(N)$ the explicit expressions are obtained for the twist element ${\cal F}$, universal ${\cal R}$-matrix and the corresponding canonical element ${\cal T}$. It is shown that the twisted Hopf algebra ${\cal U}_{\cal F} ({\bf B}^{\vee})$ is self dual. The cohomological properties of the involved Lie bialgebras are studied to justify the existence of a contraction from the Dinfeld-Jimbo quantization to the jordanian one. The construction of the twist is gene…
The recently proposed jordanian quantization of the Lie superalgebra $osp(1|2)$ due to the embedding $sl(2) \subset osp(1|2)$, is extended including odd generators into the twisting element $\cal F$. This deformation is obtained as a contraction of the quantum superalgebra ${\cal U}_{q}(osp(1|2))$.
We find the Hopf algebra $U_{g,h}$ dual to the Jordanian matrix quantum group $GL_{g,h}(2)$. As an algebra it depends only on the sum of the two parameters and is split in two subalgebras: $U'_{g,h}$ (with three generators) and $U(Z)$ (with one generator). The subalgebra $U(Z)$ is a central Hopf subalgebra of $U_{g,h}$. The subalgebra $U'_{g,h}$ is not a Hopf subalgebra and its coalgebra structure depends on both parameters. We discuss also two one-parameter special cases: $g =h$ and $g=-h$. The subalgebra $U'_{h,h}$ is a Hopf algebra and coincides with the algebra introduced by Ohn as the dua…
An explicit quantization is given of certain skew-symmetric solutions of the classical Yang-Baxter, yielding a family of $R$-matrices which generalize to higher dimensions the Jordanian $R$-matrices. Three different approaches to their construction are given: as twists of degenerations of the Shibukawa-Ueno Yang-Baxter operators on meromorphic functions; as boundary solutions of the quantum Yang-Baxter equation; via a vertex-IRF transformation from solutions to the dynamical Yang-Baxter equation.
We consider a superextension of the extended Jordanian twist, describing nonstandard quantization of anti-de-Sitter ($AdS$) superalgebra $osp(1|4)$ in the form of Hopf superalgebra. The super-Jordanian twisting function and corresponding basic coproduct formulae for the generators of $osp(1|4)$ are given in explicit form. The nonlinear transformation of the classical superalgebra basis not modifying the defining algebraic relations but simplifying coproducts and antipodes is proposed. Our physical application is to interpret the new super-Jordanian deformation of $osp(1|4)$ superalgebra as def…
The generalized Cartan type $\mathbf{S}$ Lie algebras in char 0 with the Lie bialgebra structures involved are quantized, where the Drinfel'd twist we used is proved to be a variation of the Jordanian twist. As the passage from char 0 to char p, their quantization integral forms are given. By the modular reduction and base changes, we obtain certain quantizations of the restricted universal enveloping algebra $\mathbf u(\mathbf{S}(n;\underline{1}))$ (for the Cartan type simple modular restricted Lie algebra $\mathbf{S}(n;\underline{1})$ of $\mathbf{S}$ type). They are new Hopf algebras of trun…
We construct explicit Drinfel'd twists for the generalized Cartan type $H$ Lie algebras in characteristic $0$ and obtain the corresponding quantizations and their integral forms. Via making modular reductions including modulo $p$ reduction and modulo $p$-restrictedness reduction, and base changes, we derive certain modular quantizations of the restricted universal enveloping algebra $\mathbf u(\mathbf{H}(2n;\underline{1}))$ in characteristic $p$. They are new non-pointed Hopf algebras of truncated $p$-polynomial noncommutative and noncocommutative deformation of prime-power dimension $p^{p^{2n…
By the supersymmetrization of a simple algebraic technique proposed in \cite{LuTo2017} we obtain the complete classification of all basic (nonisomorphic) quantum deformations for the orthosymplectic Lie superalgebra $\mathfrak{osp}(1|2;\mathbb{C})$ and its pseudoreal and real forms in terms of the classical $r$-matrices. In particular, we prove that pseudoreal compact form has only one quantum deformation (standart $q$-analog), and the $D=3$, $N=1$ Lorentz supersymmetry, which is the non-compact real form of $\mathfrak{osp}(1|2;\mathbb{C})$, has four different Hopf-algebraic quantum deformatio…
The quantization properties of composite peripheric twists are studied. Peripheric chains of extended twists are constructed for U(sl(N)) in order to obtain composite twists with sufficiently large carrier subalgebras. It is proved that the peripheric chains can be enlarged with additional Reshetikhin and Jordanian factors. This provides the possibility to construct new solutions to Drinfeld equations and, thus, to quantize new sets of Lie-Poisson structures. When the Jordanian additional factors are used the carrier algebras of the enlarged peripheric chains are transformed into algebras of m…
We study the behavior of the Etingof-Kazhdan quantization functors under the natural duality operations of Lie bialgebras and Hopf algebras. In particular, we prove that these functors are "compatible with duality", i.e., they commute with the operation of duality followed by replacing the coproduct by its opposite. We then show that any quantization functor with this property also commutes with the operation of taking doubles. As an application, we show that the Etingof-Kazhdan quantization of some affine Lie superalgebras coincide with their Drinfeld-Jimbo-type quantizations.
A universal $R$-matrix for the non-standard (Jordanian) quantum deformation of $sl(2,\R)$ is presented. A family of solutions of the quantum Yang--Baxter equation is obtained from some finite dimensional representations of this Lie bialgebra quantization of $sl(2,\R)$.
We use the decomposition of o(3,1)=sl(2;C)_1\oplus sl(2;C)_2 in order to describe nonstandard quantum deformation of o(3,1) linked with Jordanian deformation of sl(2;C}. Using twist quantization technique we obtain the deformed coproducts and antipodes which can be expressed in terms of real physical Lorentz generators. We describe the extension of the considered deformation of D=4 Lorentz algebra to the twist deformation of D=4 Poincare algebra with dimensionless deformation parameter.
The twisting function describing a nonstandard (super-Jordanian) quantum deformation of $osp(1|2)$ is given in explicite closed form. The quantum coproducts and universal R-matrix are presented. The non-uniqueness of the twisting function as well as two real forms of the deformed $osp(1|2)$ superalgebras are considered. One real quantum $osp(1|2)$ superalgebra is interpreted as describing the $κ$-deformation of D=1, N=1 superconformal algebra, which can be applied as a symmetry algebra of N=1 superconformal mechanics.
We develop a quantum duality principle for coisotropic subgroups of a (formal) Poisson group and its dual. Namely, starting from a quantum coisotropic subgroup (for a quantization of a given Poisson group) we provide functorial recipes to produce quantizations of the dual coisotropic subgroup (in the dual formal Poisson group). By the natural link between subgroups and homogeneous spaces, we argue a quantum duality principle for Poisson homogeneous spaces which are Poisson quotients, i.e. have at least one zero-dimensional symplectic leaf. As an application, we provide an explicit quantization…
For a complex or real algebraic group G, with g:=Lie(G), quantizations of global type are suitable Hopf algebras F_q[G] or U_q(g) over C[q,q^{-1}]. Any such quantization yields a structure of Poisson group on G, and one of Lie bialgebra on g : correspondingly, one has dual Poisson groups G^* and a dual Lie bialgebra g^*. In this context, we introduce suitable notions of quantum subgroup and of quantum homogeneous space, in three versions: weak, proper and strict (also called "flat" in the literature). The last two notions only apply to those subgroups which are coisotropic, and those homogeneo…
We introduce a general recipe to construct quantum projective homogeneous spaces, with a particular interest for the examples of the quantum Grassmannians and the quantum generalized flag varieties. Using this construction, we extend the quantum duality principle to quantum projective homogeneous spaces.
For the quantized universal enveloping algebra U_h(g_X) associated with a continuous Kac-Moody algebra g_X as in [A. Appel, F. Sala, "Quantization of continuum Kac-Moody algebras", Pure Appl. Math. Q. 16 (2020), no. 3, 439-493], we prove that a suitable formulation of the Quantum Duality Principle holds true, both in a "formal" version - i.e., applying to the original definition of U_h(g_X) as a formal QUEA over the algebra of formal series in h - and in a "polynomial" one - i.e., for a suitable polynomial form of U_h(g_X) over the algebra of Laurent polynomials in q. In both cases, the QDP st…
Let $α$ be a quadratic Poisson bivector on a vector space $V$. Then one can also consider $α$ as a quadratic Poisson bivector on the vector space $V^*[1]$. Fixed a universal deformation quantization (prediction some weights to all Kontsevich graphs [K97]), we have deformation quantization of the both algebras $S(V^*)$ and $Λ(V)$. These are graded quadratic algebras, and therefore Koszul algebras. We prove that for some universal deformation quantization, independent on $α$, these two algebras are Koszul dual. We characterize some deformation quantizations for which this theorem is true in the …
We prove that the action of the Grothendieck-Teichmüller group on the genus completed properad of (homotopy) Lie bialgebras commutes with the reversing directions involution of the latter. We also prove that every universal quantization of Lie bialgebras is homotopy equivalent to the one which commutes with the duality involution exchanging Lie bracket and Lie cobracket. The proofs are based on a new result in the theory of oriented graph complexes (which can be of independent interest) saying that the involution on an oriented graph complex that changes all directions on edges induces the ide…
Let $α$ be a polynomial Poisson bivector on a finite-dimensional vector space $V$ over $\mathbb{C}$. Then Kontsevich [K97] gives a formula for a quantization $f\star g$ of the algebra $S(V)^*$. We give a construction of an algebra with the PBW property defined from $α$ by generators and relations. Namely, we define an algebra as the quotient of the free tensor algebra $T(V^*)$ by relations $x_i\otimes x_j-x_j\otimes x_i=R_{ij}(\hbar)$ where $R_{ij}(\hbar)\in T(V^*)\otimes\hbar \mathbb{C}[[\hbar]]$, $R_{ij}=\hbar \Sym(α_{ij})+\mathcal{O}(\hbar^2)$, with one relation for each pair of $i,j=1...\d…
We introduce the notion of being cohomologically complete for objects of the derived category of sheaves of $Z[\hbar]$-modules on a topological space. Then we consider a $Z[\hbar]$-algebra satisfying some suitable conditions and prove coherency results by using the property of being cohomologically complete. We apply these results to the study of modules over deformation quantization algebroids on complex Poisson manifolds. We prove in particular that under a natural properness condition, the convolution of two coherent kernels over such algebroids is coherent. We also construct the dualizing …
In this paper we show that the T-duality transform of Bouwknegt, Evslin and Mathai applies to determine isomorphisms of certain current algebras and their associated vertex algebras on topologically distinct T-dual spacetimes compactified to circle bundles with $H$-flux.
Montonen-Olive duality implies that the categories of A-branes on the moduli spaces of Higgs bundles on a Riemann surface C for a pair of Langlands-dual groups are equivalent. We reformulate this as a statement about categories of B-branes on the quantized moduli spaces of flat connections for these groups. We show that it implies the statement of the Quantum Geometric Langlands duality with a purely imaginary ``quantum parameter'' which is proportional to the inverse of the Planck constant of the gauge theory. The ramified version of the story is also considered.
The "quantum duality principle" states that the quantization of a Lie bialgebra - via a quantum universal enveloping algebra (QUEA) - provides also a quantization of the dual Lie bialgebra (through its associated formal Poisson group) - via a quantum formal series Hopf algebra (QFSHA) - and, conversely, a QFSHA associated to a Lie bialgebra (via its associated formal Poisson group) yields a QUEA for the dual Lie bialgebra as well; more precisely, there exist functors QUEA --> QFSHA and QFSHA --> QUEA, inverse of each other, such that in either case the Lie bialgebra associated to the target ob…
The Weyl-Wigner prescription for quantization on Euclidean phase spaces makes essential use of Fourier duality. The extension of this property to more general phase spaces requires the use of Kac algebras, which provide the necessary background for the implementation of Fourier duality on general locally compact groups. Kac algebras -- and the duality they incorporate -- are consequently examined as candidates for a general quantization framework extending the usual formalism. Using as a test case the simplest non-trivial phase space, the half-plane, it is shown how the structures present in t…
We develop a quantum duality principle for coisotropic subgroups of a (formal) Poisson group and its dual: namely, starting from a quantum coisotropic subgroup (for a quantization of a given Poisson group) we provide functorial recipes to produce quantizations of the dual coisotropic subgroup (in the dual formal Poisson group). By the natural link between subgroups and homogeneous spaces, we argue a quantum duality principle for Poisson homogeneous spaces which are Poisson quotients, i.e. have at least one zero-dimensional symplectic leaf. Only bare results are presented, while detailed proofs…
We investigate the algebro-geometric structure of a novel two-parameter quantum deformation which exhibits the nature of a semidirect or cross-product algebra built upon GL(2) x GL(1), and is related to several other known examples of quantum groups. Following the R-matrix framework, we construct the L+/- functionals and address the problem of duality for this quantum group. This naturally leads to the construction of a bicovariant differential calculus that depends only on one deformation parameter, respects the cross-product structure and has interesting applications. The corresponding Jorda…
The Weyl-Wigner correspondence prescription, which makes large use of Fourier duality, is reexamined from the point of view of Kac algebras, the most general background for noncommutative Fourier analysis allowing for that property. It is shown how the standard Kac structure has to be extended in order to accommodate the physical requirements. An Abelian and a symmetric projective Kac algebras are shown to provide, in close parallel to the standard case, a new dual framework and a well-defined notion of projective Fourier duality for the group of translations on the plane. The Weyl formula ari…
We consider the problem of quantization of classical soliton integrable systems, such as the KdV hierarchy, in the framework of a general formalism of Gaudin models associated to affine Kac--Moody algebras. Our experience with the Gaudin models associated to finite-dimensional simple Lie algebras suggests that the common eigenvalues of the mutually commuting quantum Hamiltonians in a model associated to an affine algebra should be encoded by affine opers associated to the Langlands dual affine algebra. This leads us to some concrete predictions for the spectra of the quantum Hamiltonians of th…
Semigroup algebras admit certain `coherent' deformations which, in the special case of a path algebra, may associate a periodic function to an evolving path; for a particle moving freely on a straight line after an initial impulse, the wave length is that hypothesized by de Broglie's wave-particle duality. This theory leads to a model of "physical" phase space of which mathematical phase space, the cotangent bundle of configuration space, is a projection. This space is singular, quantized at the Planck level, its structure implies the existence of spin, and the spread of a packet can be descri…
We introduce and study a transfer map between ideals of the universal enveloping algebras of two members of a reductive dual pair of Lie algebras. Its definition is motivated by the approach to the real Howe duality through the theory of Capelli identities. We prove that this map provides a lower bound on the annihilators of theta lifts of representations with a fixed annihilator ideal. We also show that in the algebraic stable range, transfer respects the class of quantizations of nilpotent orbit closures. As an application, we explicitly describe quantizations of small nilpotent orbits of ge…
We develop a quantum duality principle for subgroups of a Poisson group and its dual, in two formulations. Namely, in the first one we provide functorial recipes to produce quantum coisotropic subgroups in the dual Poisson group out of any quantum subgroup (in a tautological sense) of the initial Poisson group, while in the second one similar recipes are given only starting from coisotropic subgroups. In both cases this yields a Galois-type correspondence, where a quantum coisotropic subgroup is mapped to its complementary dual; moreover, in the first formulation quantum coisotropic subgroups …
We consider a smooth Poisson affine variety with the trivial canonical bundle over complex numbers. For such a variety the deformation quantization algebra A_h enjoys the conditions of the Van den Bergh duality theorem and the corresponding dualizing module is determined by an outer automorphism of A_h intrinsic to A_h. We show how this automorphism can be expressed in terms of the modular class of the corresponding Poisson variety. We also prove that the Van den Bergh dualizing module of the deformation quantization algebra A_h is free if and only if the corresponding Poisson structure is uni…
Consider a complex symplectic manifold $X$ and the algebroid $W_X$ of quantization-deformation. For two regular holonomic modules $L_i$ ($i=0,1$) supported by smooth Lagrangian manifolds, we prove that the complex $Rhom_{W_X}(L_1,L_0)$ is constructible and perverse and dual to the complex $Rhom_{W_X}(L_0,L_1)$.
We show for any oriented surface, possibly with a boundary, how to generalize Kramers-Wannier duality to the world of quantum groups. The generalization is motivated by quantization of Poisson-Lie T-duality from the string theory. Cohomologies with quantum coefficients are defined for surfaces and their meaning is revealed. They are functorial with respect to some glueing operations and connected with q-invariants of 3-folds.
Quantum duality principle is applied to study classical limits of quantum algebras and groups. For a certain type of Hopf algebras the explicit procedure to construct both classical limits is presented. The canonical forms of quantized Lie-bialgebras are proved to be two-parametric varieties with two classical limits called dual. When considered from the point of view of quantized symmetries such varieties can have boundaries that are noncommutative and noncocommutative. In this case the quantum duality and dual limits still exist while instead of Lie bialgebra one has a pair of tangent vector…
We study modules over the algebroid stack $\W[\stx]$ of deformation quantization on a complex symplectic manifold $\stx$ and recall some results: construction of an algebra for $\star$-products, existence of (twisted) simple modules along smooth Lagrangian submanifolds, perversity of the complex of solutions for regular holonomic $\W[\stx]$-modules, finiteness and duality for the composition of ``good'' kernels. As a corollary, we get that the derived category of good $\W[\stx]$-modules with compact support is a Calabi-Yau category. We also give a conjectural Riemann-Roch type formula in this …
This paper together with the previous one (arXiv:hep-th/0604146) presents the detailed description of all quantum deformations of D=4 Lorentz algebra as Hopf algebra in terms of complex and real generators. We describe here in detail two quantum deformations of the D=4 Lorentz algebra o(3,1) obtained by twisting of the standard q-deformation U_{q}(o(3,1)). For the first twisted q-deformation an Abelian twist depending on Cartan generators of o(3,1) is used. The second example of twisting provides a quantum deformation of Cremmer-Gervais type for the Lorentz algebra. For completeness we describ…
In this paper, we initiate the study of nonassociative strict deformation quantization of C*-algebras with a torus action. We shall also present a definition of nonassociative principal torus bundles, and give a classification of these as nonassociative strict deformation quantization of ordinary principal torus bundles. We then relate this to T-duality of principal torus bundles with $H$-flux. We also show that the Octonions fit nicely into our theory.
We prove a version of Kontsevich's formality theorem for two subspaces (branes) of a vector space $X$. The result implies in particular that the Kontsevich deformation quantizations of $\mathrm{S}(X^*)$ and $\wedge(X)$ associated with a quadratic Poisson structure are Koszul dual. This answers an open question in Shoikhet's recent paper on Koszul duality in deformation quantization.
